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25
Shuffling cards and stopping times
 In Proceedings of the 43rd IEEE Conference on Decision and Control
, 1986
"... 1. Introduction. How many times must a deck of cards be shuffled until it is close to random? There is an elementary technique which often yields sharp estimates in such problems. The method is best understood through a simple example. EXAMPLE1. Top in at random shuffle. Consider the following metho ..."
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Cited by 133 (17 self)
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1. Introduction. How many times must a deck of cards be shuffled until it is close to random? There is an elementary technique which often yields sharp estimates in such problems. The method is best understood through a simple example. EXAMPLE1. Top in at random shuffle. Consider the following method of mixing a deck of cards: the top card is removed and inserted into the deck at a random position. This procedure is
Rates of Convergence for Gibbs Sampling for Variance Component Models
 Ann. Stat
, 1991
"... This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for K location parameters, with J observations each, the number of iterations required for convergence (for large K ..."
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Cited by 41 (10 self)
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This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for K location parameters, with J observations each, the number of iterations required for convergence (for large K and J) is a constant times
Approximate pvalues for local sequence alignments
 Ann. Statist
, 2000
"... Siegmund and Yakir (2000) have given an approximate pvalue when two independent, identically distributed sequences from a nite alphabet are optimally aligned based on a scoring system that rewards similarities according to a general scoring matrix and penalizes gaps (insertions and deletions). The ..."
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Cited by 40 (1 self)
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Siegmund and Yakir (2000) have given an approximate pvalue when two independent, identically distributed sequences from a nite alphabet are optimally aligned based on a scoring system that rewards similarities according to a general scoring matrix and penalizes gaps (insertions and deletions). The approximation involves an innite sequence of difculttocompute parameters. In this paper, it is shown by numerical studies that these reduce to essentially two numerically distinct parameters, which can be computed as onedimensional numerical integrals. For an arbitrary scoring matrix and afne gap penalty, this modied approximation is easily evaluated. Comparison with published numerical results show that it is reasonably accurate. Key words: local alignment, afne gap penalty, pvalue, Markov renewal theory. 1.
Rates of Convergence for Data Augmentation on Finite Sample Spaces
 Ann. Appl. Prob
, 1993
"... this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where ..."
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Cited by 28 (14 self)
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this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where
Onedimensional linear recursions with Markovdependent coefficients
, 2004
"... For a class of stationary Markovdependent sequences (ξn,ρn) ∈ R 2, we consider the random linear recursion Sn = ξn + ρnSn−1, n ∈ Z, and show that the distribution tail of its stationary solution has a power law decay. An application to random walks in random environments is discussed. MSC2000: pri ..."
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Cited by 18 (0 self)
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For a class of stationary Markovdependent sequences (ξn,ρn) ∈ R 2, we consider the random linear recursion Sn = ξn + ρnSn−1, n ∈ Z, and show that the distribution tail of its stationary solution has a power law decay. An application to random walks in random environments is discussed. MSC2000: primary 60K15; secondary 60K20, 60K37.
RENEWAL THEORY FOR FUNCTIONALS OF A MARKOV CHAIN WITH COMPACT STATE SPACE
, 2003
"... Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to ..."
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Cited by 10 (4 self)
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Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to simplify Kesten’s proof considerably.
Behavior near the extinction time in selfsimilar fragmentations I: the stable case
, 805
"... The stable fragmentation with index of selfsimilarity α ∈ [−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1 + α) −1 –stable continuum random tree below height t, for t ≥ 0. We give a detailed limiting description of the distribution of such a fragm ..."
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Cited by 7 (0 self)
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The stable fragmentation with index of selfsimilarity α ∈ [−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1 + α) −1 –stable continuum random tree below height t, for t ≥ 0. We give a detailed limiting description of the distribution of such a fragmentation, (F(t), t ≥ 0), as it approaches its time of extinction, ζ. In particular, we show that t 1/α F((ζ − t) +) converges in distribution as t → 0 to a nontrivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain sizebiased version of the law of ζ. In addition, we prove that the logarithms of the sizes of the largest fragment and last fragment to disappear, at time (ζ − t) +, rescaled by log(t), converge almost surely to the constant −1/α as t → 0.
Renewal theory in analysis of tries and strings
, 2009
"... We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes. ..."
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Cited by 4 (1 self)
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We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes.